A common misconception is that the Torah was written on Mount Sinai.
On Mount Sinai we were given the the 10 Commandments and the Oral Torah (to Moshe). The actual writing transpired before and after the Sinai experience.
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Our explanation goes by Rashi (Shemot 24:4) who says the whole Torah, until the Mt. Sinai experience, was written before the giving of the Torah. Ramban (preface to his Torah commentary), however, says that part was written after the giving of the Torah: “When Moses came down from the mountain, he wrote from the beginning of the Torah until the end of the story of the Tabernacle, and the conclusion of the Torah he wrote at the end of the fortieth year…this is according to the one who says the Torah was given scroll by scroll. But according to the one who says it was given complete, the entire thing was written in the 40th year.”
The debate regarding the writing after the giving of the Torah—whether it was scroll by scroll throughout the desert or all at once in the 40th year—appears in Bab. Talmud Gittin 60a.
